Common-limit conjecture for renormalization fixed points of one-dimensional maps

Suppose a topological class of one-dimensional maps consists of maps with one critical value, and suppose that, for each critical exponent in a sequence tending to $\infty$, limits of appropriate renormalization schemes exist and satisfy a functional fixed-point equation. Common-limit conjecture. Then there exists a topologically different class of one-dimensional maps for which limits of similar renormalization schemes exist and satisfy the same functional equation. Furthermore, as the exponent tends to $\infty$ in both classes, the fixed-point maps tend to a common limiting dynamics. The conjecture proposes a mechanism by which fixed points from distinct topological classes can share a limiting dynamics at infinite criticality; in the paper, this expectation is motivated by the observed convergence of Fibonacci circle coverings and homeomorphisms to the same limit class.